A homomorphic encryption is an original cryptography which provides an operation for encryption data. Specifically, if the encryption data is given, without using encrypted plaintext data or secret key required for decryption, a specific operation is performed on the encryption data to apply an intended operation to the encrypted plaintext data so that various types of data processing as a proxy including numerical calculation for the encryption data are allowed without losing a confidentiality of the encrypted information.
Studies on homomorphic encryption of the related are very restricted. A homomorphic encryption which provides only a characteristic of an additive group which performs only addition and subtraction or a homomorphic encryption which provides only a characteristic of a multiplicative group which performs only multiplication and division is suggested so that homomorphic encryption which stores a characteristic of a single operation is widely known. However, it is generally recognized that a design for a technique which stores two different operation groups is difficult. A technique which provides both operations is suggested based on a pairing operation defined in an elliptic curve, but the technique has a restriction that the multiplication is limited to only one time. The homomorphic encryption may be used for a statistical purpose to calculate an average of encrypted private information but the design of the homomorphic encryption which may support all arbitrary operations is still difficult problems and thus many studies have been performed in order to solve the difficult problems.
Recently, the homomorphic encryption which supports all arbitrary operations is designed by Gentry on 2009 for the first time. The technique designed by Gentry designs a somewhat homomorphic encryption (hereinafter, abbreviated as SHE) which may perform limited number of operations which is determined at a predetermined level regardless of the type of the operation and expands to a Fully homomorphic encryption (hereinafter, abbreviated as FHE) which stores the arbitrary operation based on the SHE without limitation. The FHE which is designed by the expansion technique of Gentry has most of properties such as a length of the key or an operation performance of the SHE technique which is the base of the FHE as it is. Therefore, it is recognized that the important problem is to design an efficient SHE and the SHE techniques are designed based on a hardness of the difficult problems defined in the lattice or the integer till now. The techniques which have been suggested till now provide only one bit encryption so that n ciphertexts are generated through 1 bit encryptions.
Recently, the studies on the homomorphic encryption design focus to realize a length of the public key of the suggested techniques. Such studies are very appropriate and necessary because the known techniques use very large public keys. The homomorphic encryption mainly has a cloud service or a big data service where large quantity data is stored in a server in an actual application environment as a main application target so that a technology which reduces not only a length of the public key which uses a fixed spatial resource but also a size of the encryption data is important and necessary in order to be used in an actual application environment. Actually, the public key has a fixed length. However, as a size of data which is stored by the user is increased, a service provider needs to pay for a storage space in proportional to the increasing size of data. As a result, in order to use the homomorphic encryption as a primitive for acting as a proxy in stable and efficient processing of the encryption data, it is very important to reduce the size of the encryption data generated when the same data is encrypted.